Zeros of meromorphic solutions of second order linear differential equations (Q1124017)

Given the equation (*) \(y''+H(z)y=0\), let \(H=P/Q\) be rational (P,Q are polynomials), and let \(n=\deg P-\deg Q\). The main result of the paper is that if \(f_ 1\), \(f_ 2\) are linearly independent meromorphic solutions of (*) such that (1) \(f_ 1f_ 2\) is transcendental, and (2) the number of nonreal zeros of \(f_ 1\) and \(f_ 2\) in the disc\(| z| \leq r\) is \(o(r^{(n+2)/2})\), as \(r\to \infty\), then n must be 0. By means of an example, the authors show that the conclusion \(n=0\) may not be replaced by \(H=\)constant, as in the case where H(z) is a polynomial, which was treated by \textit{S. Hellerstein}, \textit{L. C. Shen} and \textit{J. Williamson} [Trans. Am. Math. Soc. 285, 759-776 (1984; Zbl 0549.30021)] and \textit{G. Gundersen} [Ann. Acad. Sci. Fenn. Ser. A I 11, 275-294 (1986; Zbl 0607.34007)]. The authors also study asymptotic estimates of the Nevanlinna functions of a meromorphic solution of (*) and deduce the precise possible values of the deficiencies of zeros. However, the proof of the main result is not based on the Nevanlinna theory, but on \textit{E. Hille's} ``Oscillation theorems in the complex domain'' [see, e.g., Lectures on ordinary differential equations (1969; Zbl 0179.403)]. There are also several references to a related paper by \textit{S. Bank} and \textit{I. Laine} [Comment. Math. Helv. 58, 656-677 (1983; Zbl 0532.34008)].